A000620 Number of monosubstituted alkanes C(n-1)H(2n-1)-X with n-1 carbon atoms that are stereoisomers. (Formerly M1642 N0642)

0, 0, 0, 0, 2, 6, 20, 60, 176, 512, 1488, 4326, 12648, 37186, 109980, 327216, 979020, 2944414, 8897732, 27005290, 82288516, 251650788, 772127678, 2376238138, 7333188770, 22688297950, 70360977228, 218678818026, 681017928476, 2124840874610, 6641336507270, 20791999731518

Offset

1,5

Comments

Also number of monosubstituted alkanes C(n)H(2n+1)-X of the form R-CH2-X (primary) that are stereoisomers.

Let the entries in the nine columns of Blair and Henze's Table I (JACS 54 (1932), p. 1098) be denoted by Ps(n), Pn(n), Ss(n), Sn(n), Ts(n), Tn(n), As(n), An(n), T(n) respectively (here P = Primary, S = Secondary, T = Tertiary, s = stereoisomers, n = non-stereoisomers and the last column T(n) gives total).

Then Ps (and As) = this sequence, Pn (and An, Sn) = A000621, Ss = A000622, Ts = A000623, Tn = A000624, T = A000625. Recurrences generating these sequences are given in the Maple program in A000620.

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..1930

Jean-François Alcover, Mathematica program

C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1106.

C. M. Blair and H. R. Henze, The number of stereoisomeric and non-stereoisomeric mono-substitution products of the paraffins, J. Amer. Chem. Soc., 54 (3) (1932), 1098-1105. (Annotated scanned copy)

Formula

See Maple program for recurrences for this sequence and A000621-A000625.

a(n) ~ c * b^n / n^(3/2), where b = 3.287112055584474991259... (see A239803), c = 0.105352133282419523497... (see A239805). - Vaclav Kotesovec, Mar 27 2014

Maple

# Blair and Henze's recurrences for A000620-A000625 (see comments lines for relationship between the sequences and their symbols).
Ps := [0, 0, 0]; Pn := [1, 1, 1]; Ss := [0, 0, 0]; Sn := [0, 0, 1]; Ts := [0, 0, 0]; Tn := [0, 0, 0]; As := [0, 0, 0]; An := [1, 1, 2]; T := [1, 1, 2];
for n from 4 to 60 do Ps := [op(Ps), As[n-1]]; Pn := [op(Pn), An[n-1]]; t1 := add( 2*T[n-1-j]*T[j], j=1..floor((n-2)/2) ); if n mod 2 = 1 then i := (n-1)/2; t1 := t1+T[i]^2-An[i]; fi; Ss := [op(Ss), t1];
t2 := 0; if n mod 2 = 1 then i := (n-1)/2; t2 := An[i]; fi; Sn := [op(Sn), t2]; t3 := 0; for i from 1 to (n-1)/3 do for j from i+1 to (n-2)/2 do k := n-1-i-j; if j<k then t3 := t3+2*T[i]*T[j]*T[k]; fi; od: od:
t4 := 0; t5 := 0; for i from 1 to (n-2)/2 do j := n-1-2*i; if j > 0 and i <> j then t4 := t4+(T[i]^2-An[i])*T[j]+An[i]*As[j]; t5 := t5+An[i]*An[j]; fi; od; t6 := 0; t7 := 0; if n mod 3 = 1 then i := (n-1)/3; t6 := (2*T[i]+T[i]^3)/3-An[i]^2; t7 := An[i]^2; fi;
Ts := [op(Ts), t3+t4+t6]; Tn := [op(Tn), t5+t7]; As := [op(As), Ps[n]+Ss[n]+Ts[n]]; An := [op(An), Pn[n]+Sn[n]+Tn[n]]; T := [op(T), As[n]+An[n]]; od: Ps; Pn; Ss; Ts; Tn; T;

Mathematica

(* See links *)

Crossrefs

Cf. A000621, A000622, A000623, A000624, A000625, A239803, A239805.

Sequence in context: A358301 A361732 A005628 * A081251 A134293 A363107

Adjacent sequences: A000617 A000618 A000619 * A000621 A000622 A000623

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Additional comments from Bruce Corrigan, Nov 04 2002

Status

approved